This paper is a survey of the \(K\)-theoretic generalization of class field theory. For a field \(K\), let \(K^{\text{ab}}\) be a maximal abelian extension of \(K\), that is, the union of all finite abelian extensions of \(K\) in a fixed algebraic closure of \(K\). The classical local (resp. global) class field theory says that if \(K\) is a finite extension of the \(p\)-adic (resp. rational) number field \(\mathbb{Q}_p\) (resp. \(\mathbb{Q}\)), the Galois group \(\text{Gal} (K^{\text{ab}}/ K)\) is approximated by the multiplicative group \(K^\times\) (resp. the idele class group \(C_K\)), and via this approximation, we can obtain knowledge on abelian extensions of \(K\).
In §1 (resp. §2), we give a \(K\)-theoretic generalization of the classical local (resp. global) class field theory. There finite extensions of \(\mathbb{Q}_p\) (resp. \(\mathbb{Q}\)) are replaced by “higher dimensional local fields” (resp. finitely generated fields over prime fields), and the group \(K^\times\) (resp. \(C_K)\) is replaced by Milnor’s \(K\)-group \(K_n^M (K)\) (resp. by the \(K_n^M\)-idele class group), where \(n\) is the “dimension” of \(K\).
In §3, we discuss some other aspects of generalizations of local class field theory. In §4, we discuss generalizations of the classical ramification theory to higher dimensional schemes.
For the entire collection see [Zbl 0741.00019].